Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality

Details Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality

2011-07-31

arxiv.org/abs/1108.0132v1

Mathematics

Classical Analysis and ODEs

12 pages, 14 references

Scientific paper

We introduce the -1 dual Hahn polynomials through an appropriate $q \to -1$ limit of the dual q-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical orthogonal polynomials of the Askey scheme, the -1 dual Hahn polynomials do not exhibit the Leonard duality property. Instead, these polynomials satisfy a 4-th order difference eigenvalue equation and thus possess a bispectrality property. The corresponding generalized Leonard pair consists of two matrices $A,B$ each of size $N+1 \times N+1$. In the eigenbasis where the matrix $A$ is diagonal, the matrix $B$ is 3-diagonal; but in the eigenbasis where the matrix $B$ is diagonal, the matrix $A$ is 5-diagonal.

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